Partial Derivative Calculator

What a partial derivative means

A partial derivative measures how a multivariable function changes as one variable moves while the other variables are held fixed. For a function f(x, y, z), the derivative ∂f/∂x treats y and z as constants. This is the core rule behind the calculator: differentiate only the selected variable, carry all other variable factors along unchanged, and drop any term that does not contain the selected variable.

For a polynomial term, the power rule still applies:

∂/∂x (a * x^m * y^n * z^p) = a * m * x^(m-1) * y^n * z^p

The same pattern works for y or z; only the chosen variable supplies the exponent multiplier.

Worked example

For f(x, y, z) = 3x^2y + 4xy^3 - 5z + 7:

If you evaluate ∂f/∂x = 6xy + 4y^3 at (x, y, z) = (2, 3, 1), the value is 6 * 2 * 3 + 4 * 3^3 = 144. Notice that z does not appear in this derivative, so changing z would not change this specific value.

Supported syntax

The calculator is intentionally deterministic. It supports polynomial-style terms in x, y and z: implicit products such as 3x^2y, explicit products such as x*y, decimal coefficients such as 2.5z, constants and plus or minus signs. It does not process trig functions, logarithms, parentheses, quotients or arbitrary scripts. For those broader symbolic cases, use the result as a quick polynomial checker rather than a full computer algebra system.

Common pitfalls

• Forgetting the constant-variable rule. In ∂/∂x, the term 4xy^3 becomes 4y^3, not 12xy^2.
• Dropping carried factors. 3x^2y becomes 6xy; the y remains because it is a constant multiplier.
• Confusing a partial derivative with a total derivative. A partial derivative changes one input at a time. A total derivative also accounts for how variables depend on each other.
• Expecting unsupported functions. Expressions like sin(xy) or (x + y)^2 need expansion or a symbolic algebra tool before this calculator can process them.